Step 1 :The initial angular velocity of the tire is \(3.80 \, \mathrm{rad/s}\) in the counterclockwise direction. This is considered as positive.
Step 2 :The tire is then spun in the opposite (clockwise) direction at the same speed, so the final angular velocity is \(-3.80 \, \mathrm{rad/s}\).
Step 3 :The change in angular velocity \(\Delta \omega\) is given by the final angular velocity minus the initial angular velocity. So, \(\Delta \omega = -3.80 \, \mathrm{rad/s} - 3.80 \, \mathrm{rad/s} = -7.60 \, \mathrm{rad/s}\).
Step 4 :The average angular acceleration \(\alpha_{\mathrm{av}}\) is given by the change in angular velocity divided by the change in time. The change in time \(\Delta t\) is given as \(2.50 \, \mathrm{s}\). So, \(\alpha_{\mathrm{av}} = \frac{-7.60 \, \mathrm{rad/s}}{2.50 \, \mathrm{s}} = -3.04 \, \mathrm{rad/s^2}\).
Step 5 :Final Answer: The change in the tire's angular velocity \(\Delta \omega\) is \(\boxed{-7.60 \, \mathrm{rad/s}}\) and the tire's average angular acceleration \(\alpha_{\mathrm{av}}\) is \(\boxed{-3.04 \, \mathrm{rad/s^2}}\).